Cluster Alphabets from Generalized Worldsheets: A Geometric Approach to Finite Types

TL;DR

This paper introduces a geometric method to systematically derive cluster alphabets of finite types using generalized worldsheets, simplifying the known sets through a new gauge choice and evolution of cross ratios.

Contribution

It presents a novel geometric construction of cluster alphabets for finite types, improving simplicity and understanding over previous methods.

Findings

Derived simpler cluster alphabets via geometric realization.

Used Y-system equations to evolve cross ratios.

Provided a systematic derivation approach for finite type cluster algebras.

Abstract

We provide a systematic derivation of cluster alphabets of finite types. The construction is based on a geometric realization of the generalized worldsheets by gluing and folding a pair of polygons. The cross ratios of the worldsheet z variables are evolved using the Y-system equations. By a new gauge choice, we obtain a simpler set of cluster alphabets than the known ones.